
Q1 Determine order and degree(if defined) of differential equation \begin{align} \frac{d^4y}{dx^4}\;+\;\sin(y^m)\;=0\end{align}
Ans: \begin{align} \frac{d^4y}{dx^4}\;+\;\sin(y^m)\;=0 \end{align}
\begin{align} \Rightarrow y^{m\;'}+\;\sin(y^m)\;=0 \end{align}
The highest order derivative present in the differential equation is y^{m '}. Therefore, its order is four.
The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined.
Q2 Determine order and degree(if defined) of differential equation y' + 5y = 0
Ans: The given differential equation is:
y' + 5y = 0
The highest order derivative present in the differential equation isy'. Therefore, its order is one.
It is a polynomial equation in y'. The highest power raised to y' is 1. Hence, its degree is one.
Q3 Determine order and degree(if defined) of differential equation \begin{align}\left(\frac{ds}{dt}\right)^4\;+\;3s\frac{d^2s}{dt^2}\;=\;0\end{align}
Ans: \begin{align}\left(\frac{ds}{dt}\right)^4\;+\;3s\frac{d^2s}{dt^2}\;=\;0\end{align}
The highest order derivative present in the given differential equation is\begin{align}\frac{d^2s}{dt^2}.\end{align}
Therefore, its order is two. It is a polynomial equation in
\begin{align}\frac{d^2s}{dt^2} and \frac{ds}{dt}.\end{align}
The power raised to is 1. \begin{align} \frac{d^2s}{dt^2} \end{align}
Q4 Determine order and degree(if defined) of differential equation
\begin{align}\left(\frac{d^2y}{dx^2}\right)^2\;+\;cos\left(\frac{dy}{dx}\right)\;=\;0\end{align}
Ans: \begin{align}\left(\frac{d^2y}{dx^2}\right)^2\;+\;cos\left(\frac{dy}{dx}\right)\;=\;0\end{align}
The highest order derivative present in the given differential equation is \begin{align}\frac{d^2y}{dx^2}.\end{align}
Therefore, its order is 2. The given differential equation is not a polynomial equation in its derivatives.
Hence, its degree is not defined.
Q5 Determine order and degree(if defined) of differential equation \begin{align}\frac{d^2y}{dx^2}=\cos3x + sin3x\end{align}
Ans: \begin{align}\frac{d^2y}{dx^2}=\cos3x + sin3x\end{align}
\begin{align}\Rightarrow\frac{d^2y}{dx^2}  \cos3x  sin3x = 0\end{align}
The highest order derivative present in the differential equation is\begin{align}\frac{d^2y}{dx^2}.\end{align}
Therefore, its order is two.It is a polynomial equation in \begin{align}\frac{d^2y}{dx^2}\end{align}
and the power raised to is 1.
\begin{align}\frac{d^2y}{dx^2}\end{align}
Hence, its degree is one.
Q6 Determine order and degree(if defined) of differential equation (y^{m})^{2} + (y^{n})^{3} + (y')^{4} + y^{5} =0
Ans: (y^{m})^{2} + (y^{n})^{3} + (y')^{4} + y^{5} =0
The highest order derivative present in the differential equation isy^{m}. Therefore, its order is three.
The given differential equation is a polynomial equation in y^{m , }y^{n , y'.}
The highest power raised to y^{m }is 2. Hence, its degree is 2.
Q7 Determine order and degree(if defined) of differential equation y^{m} + 2y^{n} + y' =0
Ans: The highest order derivative present in the differential equation is y^{m}. Therefore, its order is three.
It is a polynomial equation in y^{m },^{ }y^{n }and y' . The highest power raised to y^{m} is 1. Hence, its degree is 1.
Q8 Determine order and degree(if defined) of differential y^{'} + y =e^{x}
Ans: y^{'} + y =e^{x}
y^{'} + y  e^{x }=0
The highest order derivative present in the differential equation is y^{'}. Therefore, its order is one.
The given differential equation is a polynomial equation in y^{' }and the highest power raised to y^{' }is one. Hence, its degree is one.
Q9 Determine order and degree(if defined) of differential equation y^{n} + (y')^{2} + 2y =0
Ans: y^{n} + (y')^{2} + 2y =0
The highest order derivative present in the differential equation is y^{n}. Therefore, its order is two.
The given differential equation is a polynomial equation in y^{n }and y'^{ }and the highest power raised to y^{n }is one.
Hence, its degree is one.
Q10 Determine order and degree(if defined) of differential equation y^{n} + 2y^{'} + siny = 0
Ans: y^{n} + 2y^{'} + siny = 0
The highest order derivative present in the differential equation is y^{n}. Therefore, its order is two.
This is a polynomial equation in y^{n }and y' and the highest power raised to y^{n }is one. Hence, its degree is one.
Q11 The degree of the differential equation
\begin{align}\left(\frac{d^2y}{dx^2}\right)^3\;+ \left(\frac{dy}{dx}\right)^2+\;sin\left(\frac{dy}{dx}\right)\;+ 1=\;0\end{align}
is (A) 3 (B) 2 (C) 1 (D) not defined
Ans: \begin{align}\left(\frac{d^2y}{dx^2}\right)^3\;+ \left(\frac{dy}{dx}\right)^2+\;sin\left(\frac{dy}{dx}\right)\;+ 1=\;0\end{align}
The given differential equation is not a polynomial equation in its derivatives. Therefore, its degree is not defined.
Hence, the correct answer is D.
Q12 The order of the differential equation
\begin{align}2x^2\frac{d^2y}{dx^2}\; \;3\frac{dy}{dx}\;+ y=\;0\end{align}
is (A) 2 (B) 1 (C) 0 (D) not defined
Ans: \begin{align}2x^2\frac{d^2y}{dx^2}\; \;3\frac{dy}{dx}\;+ y=\;0\end{align}
The highest order derivative present in the given differential equation is
\begin{align}\frac{d^2y}{dx^2}\end{align}
Therefore, its order is two.
Hence, the correct answer is A.